# Random Variable

**Random Variable:**

In an experiment of tossing a pair of coin we may be interested in obtaining the probabilities of getting 0,1 or 2 heads which are possible events. We consider a variable *X *which hold these possible values 0,1 or 2. This variable is a real valued function defined over sample space whose range is non empty set of real numbers is called random variable. i.e, A random variable is a function *X*(*w*) with domain *S *and range (−infinite,infinite) such that for every real number *a*, the event [*e *: *X*(*e*) ≤ *a*].

**Mathematical Expectation or Expected Value of a Random Variable:**

The expected value of a discrete random variable is a weighted average of all possible values of the random variable, where the weights are the probabilities associated with the corresponding values. For discret random variable. The expected value of a discrete random variable X with probability mass function (p.m.f) f (x) is given below:

For continuous random variable. The mathematical expression for computing the expected value of a continuous random varriable X with probability density function (p.d.f.) f(x) is, however, as follows:

which is defined as μ′_{r }the rth moment (about origin) of the probability distribution. Thus

**Properties of Expectation:**

**1**. Addition theorem: If X and Y are random variables, then E(aX bY) = aE (X) bE (Y) and we can generalise above result as if all the expectations exist.

**2**. Multiplication theorem: If X andY are random variables, then E(XY) = E(X) E(Y) provided X and Y are independent and we can generalise above result as if all the expectations exist.

**3**. Expectation of a linear Combination of Random Variable: Let X_{1},X_{2}, . . . ,Xn be any n random variables and if a_{1},a_{2}, . . . ,a_{n} are any n constants, then

**4**. If X ≥ 0 then E (X) ≥ 0.

**5**. If X, Y are two random variable s.t.Y ≤ X, then E(Y) ≤ E(X)

**6**. | E(X)| ≤ E(|X|)